Initial value of Newton Raphson Method

I am currently studying Newton-Raphson Method. I feel that I understand the concept of it. Somehow, I am facing some question in my head about how to actually apply it.

The questions that I have are below - How should I decide the first initial value? - How should I find all the roots on x-axis?, how should I set the ranges to find them separately?

Please, let me here your expertise. I am sorry if I have tagged my question in the wrong places. Thank you.

• To find the other roots, after your first approximation, you can divide the original equation by the factor you found, of course for algebraic equations only. – tpb261 Jul 4 '14 at 11:01
• Can you please take an example for me? – user122358 Jul 4 '14 at 11:33
• Let's say the equation is $x^3+3x^2+3x+1=0$ :D. One root is found to be -1. Then divide the original expression by $x+1$ to get $x^2+2x+1=0$. By observation, you can see that x=-1 is a triple root, but the program can't so, as a general rule, we have to divide the original expression by the factor. – tpb261 Jul 4 '14 at 11:55

It sometimes helps if you can isolate the roots in intervals. If you can find $a < b$ such that one of $f(a)$ and $f(b)$ is positive and the other negative (and your function is continuous), you know that there is a solution somewhere in the interval $(a,b)$. If in addition $f$ is monotone on this interval, you know that there is only one solution there.
• Preferably (to me, at least !), start at a point $x_0$ such that $f(x_0) f''(x_0)>0$ – Claude Leibovici Jul 4 '14 at 7:57